- Two prime numbers do not have any common factor other than \(1.\) One \((1)\) is the only common factor of all the prime numbers.
- GCF of two prime numbers is the highest common factor between both the prime numbers.
- So, GCF of two prime numbers is always 1.

**For example: **Find the GCF of the two prime numbers, \(13\) and \(17\).

Factors of \(13=1×13\)

Factors of \(17=1×17\)

Only 1 is the common factor of \(13\) and \(17\).

Thus, GCF of \(13\) and \(17\) is \(1.\)

- The co-prime numbers are the set of two or more numbers which has only one (1) as the common factor.
- Thus, GCF of co-prime numbers is always one (1).

**For example:**

Find the GCF of the co-prime numbers, 14 and 15.

Factors of 14 = 1 × 2 × 7

Factors of 15 = 1 × 3 × 5

Only 1 is the common factor of 14 and 15.

Thus, GCF of 14 and 15 is 1.

- GCF or greatest common factor is also known as the greatest common divisor (GCD).
- GCF of two numbers is the largest factor which is common in those numbers.
- We can understand GCF with the help of distributive property as discussed below.

\((24+60)=12(2+5)\)

- Here, \(12\) is the greatest common factor (GCF) of \(24\) and \(60\).
- \(2\) and \(5\) are relatively prime or co-prime numbers.

- Consider the two whole numbers, \(18\) and \(24\).

Factors of 18 = 2, 3, **6**, 9

Factors of 24 = 2, 3, 4, **6**, 8, 12

- Here, \(6\) is the greatest common factor of both the whole numbers, \(18\) and \(24\).
- Through distributive property, we can write them as shown.

- If we take \(3\) as the GCF of \(18\) and \(24\) then through distributive property, it can be written as: \((18+24)=3(6+8)\)
- Here, \(6\) and \(8\) are not relatively prime because they have the number \(2\) as the common factor.

Thus, \(3\) is not the GCF of \(18\) and \(24\).

A \(3\)

B \(9\)

C \(7\)

D \(18\)

- GCF or greatest common factor is also known as the greatest common divisor (GCD).
- GCF of two numbers is the largest factor which is common in those numbers.
- To find the GCF of the given numbers, we should follow the following steps:

- First, factorize them with the help of a factor tree to find out their prime factors.
- Choose the common prime factors.
- Multiply those common factors together and we get the GCF of the given numbers.

**For example:**

- Consider the two numbers, \(30\) and \(36\).

**Step-1** Find the prime factors of \(30\) and \(36\) with the help of the factor trees as shown.

Prime factors of \(30=2,\;3\) and \(5\)

Prime factors of \(36=2,\;2,\;3\) and \(3\)

**Step-2** The common prime factors \(=2\) and \(3\)

**Step-3 **G.C.F. \(=2×3=6\)

Thus, the GCF of \(30\) and \(36=6\)

Prime factors of \(30=2,\;3\) and \(5\)

Prime factors of \(36=2,\;2,\;3\) and \(3\)

**Step-2** The common prime factors \(=2\) and \(3\)

**Step-3 **G.C.F. \(=2×3=6\)

Thus, the GCF of \(30\) and \(36=6\)

- We know how to find the GCF but we do not know how to use its concept to solve the real life problems.
- To solve the real world problems, we should know what exactly is being asked in the problem.

The problems on GCF may ask us to:

- Split the things into smaller sections/parts.
- Equally distribute two or more items into their largest groups.
- Arrange something into rows, columns or groups.

**For example:**

Mr. Thompson has \(40\) chocolates and \(60\) candies. Find the largest number of students among whom she can distribute these items evenly.

- In this example, the items are to be distributed equally among the largest number of students, so we have to find the GCF of the two numbers.

A \(18\;\text{inches}\)

B \(16\;\text{inches}\)

C \(17\;\text{inches}\)

D \(20\;\text{inches}\)